On no-slip boundary conditions for the incompressible Navier-Stokes equations

Abstract

Two local implementations of no-slip boundary conditions are investigated for both the vorticity-streamfunction and momentum-pressure formulations of the time-dependent planar incompressible Navier-Stokes equations, as applied to barotropic ocean circulation modelling. The objective is to determine the extent to which the local accuracy and numerical consistency of these conditions affects the global solution. The effects of a non-local implementation of no-slip conditions for the vorticity-streamfunction equations are also studied. In all cases, boundary condition effects are measured by comparing time-averaged dynamics of turbulent solutions of numerical models based on the two formulations. In the model interior, the energy and enstrophy conserving Arakawa Jacobian is used for the vorticity-streamfunction equations while an extension of the energy and potential enstrophy conserving Arakawa and Lamb finite difference scheme is used for the momentum-pressure equations. Numerical experiments performed with a non-linear model similar to Bryan's barotropic ocean reveal no significant differences between the time-averaged solutions obtained with either of the two formulations, with each using either of the two local boundary conditions. A simple one-dimensional analogue of the vorticity-streamfunction equations is solved algebraically to explain the experimental results. A similar analogue suggests that an apparent inconsistency in the no-slip boundary conditions within the Cox stratified, primitive equation, ocean circulation model should not affect the accuracy or convergence of the global solution. © 1988.